Traveling capillary waves on the boundary of a disc

TL;DR

This paper introduces a new class of traveling wave solutions on the boundary of a 2D ideal fluid droplet, using Hamiltonian variables and conformal mapping, extending previous models to bounded fluids.

Contribution

It extends the theory of potential flow and surface waves to bounded droplets, deriving a pseudodifferential equation similar to the Babenko equation for these new solutions.

Findings

Discovery of a new class of traveling wave solutions

Derivation of a pseudodifferential equation for bounded fluid droplets

Extension of existing models to finite, bounded fluid domains

Abstract

We find a new class of solutions that are traveling waves on the boundary of two--dimensional droplet of ideal fluid. We assume that the free surface is subject only to the force of surface tension, and the fluid flow is potential. We use the canonical Hamiltonian variables discovered in the work of V.E. Zakharov in 1968, and conformally map lower complex plane to the interior of a fluid droplet. We write the equations in the form originally discovered in A.I. Dyachenko in (2001) for infinitely deep water, and adapted to bounded fluid in the work of S.A. Dyachenko (2019). The new class of solutions satisfies a pseudodifferential equation which is similar to the Babenko equation for the Stokes wave.